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Classical theory of crystals states that a medium to be consideredhomogeneous must satisfy the following requirements: a) the dimension of theelementary cell must be much smaller than the incident wavelength; b) thesample must contain a large number of elementary cells, i.e. it must bemacroscopic with respect to wavelength. Under these conditions, macroscopicquantities can be introduced in order to describe the optical response of themedium. We analytically demonstrate that for a symmetric elementary cell thoserequirements can be relaxed, and it is possible to assign a permittivity and apermeability to a composite structure, even if the metamaterial cannot beconsidered homogeneous under the requirements stated above. However, theeffective permittivity and permeability in some cases may give rise tounphysical, effective behaviors inside the medium, notwithstanding the factthat they satisfy requirements like being Kramers-Kronig pairs, for example,and are consistent with all the linear properties outside the structure (i.e.reflection, transmission, and absorption at all frequencies). In somesituations the medium is assigned a magnetic response even though the medium isnot magnetically active. In particular, we demonstrate that the homogenizationprocedure can lead to a medium that locally violates the second principle ofthermodynamics. We also show that, in the non-homogeneous regime, it is notpossible to describe the nonlinear behavior of the structure using an effectiveparameters approach, despite the possibility to assign an effective linearrefractive index.

Homogenization procedure for a metamaterial and local violation of the second principle of thermodynamics